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Theorem neindisj 22842
Description: Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neindisj (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}))) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)

Proof of Theorem neindisj
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
21clsss3 22784 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑋)
32sseld 3982 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ 𝑃 ∈ 𝑋))
43impr 454 . . . . 5 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ 𝑃 ∈ 𝑋)
51isneip 22830 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
64, 5syldan 590 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
7 3anass 1094 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†)) ↔ (𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))))
81clsndisj 22800 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ (𝑔 ∈ 𝐽 ∧ 𝑃 ∈ 𝑔)) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
97, 8sylanbr 581 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ (𝑔 ∈ 𝐽 ∧ 𝑃 ∈ 𝑔)) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
109anassrs 467 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑔 ∈ 𝐽) ∧ 𝑃 ∈ 𝑔) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
1110adantllr 716 . . . . . . . 8 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ 𝑃 ∈ 𝑔) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
1211adantrr 714 . . . . . . 7 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑔 ∩ 𝑆) β‰  βˆ…)
13 ssdisj 4460 . . . . . . . . . 10 ((𝑔 βŠ† 𝑁 ∧ (𝑁 ∩ 𝑆) = βˆ…) β†’ (𝑔 ∩ 𝑆) = βˆ…)
1413ex 412 . . . . . . . . 9 (𝑔 βŠ† 𝑁 β†’ ((𝑁 ∩ 𝑆) = βˆ… β†’ (𝑔 ∩ 𝑆) = βˆ…))
1514necon3d 2960 . . . . . . . 8 (𝑔 βŠ† 𝑁 β†’ ((𝑔 ∩ 𝑆) β‰  βˆ… β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
1615ad2antll 726 . . . . . . 7 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ ((𝑔 ∩ 𝑆) β‰  βˆ… β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
1712, 16mpd 15 . . . . . 6 (((((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) ∧ 𝑔 ∈ 𝐽) ∧ (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
1817rexlimdva2 3156 . . . . 5 (((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) ∧ 𝑁 βŠ† 𝑋) β†’ (βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
1918expimpd 453 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ ((𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑃 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
206, 19sylbid 239 . . 3 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑃 ∈ ((clsβ€˜π½)β€˜π‘†))) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))
2120exp32 420 . 2 (𝐽 ∈ Top β†’ (𝑆 βŠ† 𝑋 β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…))))
2221imp43 427 1 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}))) β†’ (𝑁 ∩ 𝑆) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22616  clsccl 22743  neicnei 22822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22617  df-cld 22744  df-ntr 22745  df-cls 22746  df-nei 22823
This theorem is referenced by:  clslp  22873  flimclslem  23709  utop3cls  23977
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